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**16**

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**2**answers

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### Maximum of a function of one variable

Let $D$ be a circular quadrilateral (that is a Jordan region whose boundary consists of 4 arcs
of circles all orthogonal to the unit circle) whose interior angles are all equal to 0, the vertices lie ...

**7**

votes

**5**answers

2k views

### Generalizations of Belyi's theorem

Belyi's theorem states that the following properties of a nonsingular projective algebraic curve $X$ are equivalent:
1) $X$ is defined over $\overline{\mathbb{Q}};$
2) There exists a meromorphic ...

**14**

votes

**5**answers

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### can you fool SnapPea?

A while back I thought I had some simple knots that "fooled" SnapPea. But I no longer remember those examples, if I ever had them to begin with.
What I'm looking for is a non-hyperbolizable knot ...

**14**

votes

**3**answers

694 views

### The number of cusps of higher-dimensional hyperbolic manifolds

Suppose $n$ is an integer greater than 3. Sometimes ago I heard somewhere that it is still not known if there exist complete finite-volume hyperbolic $n$-manifolds having exactly one cusp.
Could ...

**9**

votes

**3**answers

612 views

### Is the cut locus of a generic point in a hyperbolic manifold a generic polyhedron?

Let $p\in M$ be a point in a closed riemannian manifold $M$. Recall that the cut locus of $p$ is the subset of $M$ consisting of all points that are connected to $p$ by at least 2 distance-minimizing ...

**8**

votes

**1**answer

439 views

### Geodesics on the twisted pseudosphere (Dini's surface)

I wonder how difficult it is to compute geodesics on Dini's Surface,
a twisted pseudosphere?
Here is one parametrization, from
Alfred Gray's Modern Differential Geometry of Curves and Surfaces, ...

**10**

votes

**3**answers

421 views

### Torsion in cuspidal cohomology

Following Lemma 2.7 from Vogtmann's Rational Homology of Bianchi Groups, I want to define cuspidal cohomology as $$H_{\mathrm{cusp}}(M)=\frac{H_1(M)}{i_*(H_1(\partial M))}$$ where $i:\partial M\to M$ ...

**7**

votes

**1**answer

320 views

### Fundamental group of an hyperbolic $4$-manifold

Good afternoon everyone,
I have a very general question about hyperbolic manifolds and their fundamental groups in high dimension (at least $4$). If the theory of surfaces and $3$-manifolds provide ...

**7**

votes

**2**answers

754 views

### Poincaré disk model: is this locus a known curve?

Please, consider a line segment $AB$ in the Poincaré disk model. Now, consider the set $S$ of all point $P$ in the disk such that the angle $\angle APB$ is constant.
Question: is $S$ a known curve?
...

**6**

votes

**1**answer

57 views

### Distances between boundaries in a hyperbolic pants

Consider a hyperbolic pair of pants with totally-geodesic boundaries of lengths $l_i$ for $i \in \{1,2,3\}$. For any two distinct boundary components, is the length of the shortest geodesic connecting ...

**5**

votes

**1**answer

262 views

### Fixed points on boundary of hyperbolic group

Let G be a word-hyperbolic group with torsion and let ∂G be its boundary. Do there exist criteria that imply that all non-trivial finite order elements of G act fixed-point freely on ∂G?

**2**

votes

**0**answers

124 views

### How to pick out harmonics based on boundary conditions?

(..this is almost a continuation of my last question (which got closed!)...) Let me first rewrite one of the main results of this paper, http://calvino.polito.it/~camporesi/JMP94.pdf in a coordinate ...

**2**

votes

**3**answers

578 views

### Books that discuss spectral graph theory and its connection to eigenvalue problems in hyperbolic geometry

Hello,
Could you name a couple of books or downloadable lecture notes that discuss spectral graph theory and its connection to spectral problems in hyperbolic Riemann surfaces ? You could also ...